Shape formation is one of the most thoroughly studied problems in programmable matter and swarm robotics. However, in many models, the class of shapes that can be formed is highly restricted due to the particles' limited memory. In the hybrid model, an active agent with the computational power of a deterministic finite automaton can form shapes by lifting and placing passive tiles on the triangular lattice. We study the shape reconfiguration problem where the agent additionally has the ability to distinguish so-called target nodes from non-target nodes and needs to form a target shape from the initial tile configuration. We present a worst-case optimal $O(mn)$ algorithm for simply connected target shapes, where $m$ is the initial number of unoccupied target nodes and $n$ is the total number of tiles. Furthermore, we show how an agent can reconfigure a large class of target shapes with holes in $O(n^4)$ steps.

翻译：形状构建是可编程物质与群体机器人领域中被广泛研究的核心问题之一。然而，在许多模型中，由于粒子存储容量有限，可构建的形状类别受到极大限制。在混合模型中，一个具备确定性有限自动机计算能力的主动智能体，能够通过在三角晶格上拾取与放置被动瓦片来构建形状。本文研究形状重构问题，其中智能体额外具备区分目标节点与非目标节点的能力，并需要从初始瓦片配置中构建目标形状。针对单连通目标形状，我们提出了一种最坏情况下最优的 $O(mn)$ 算法，其中 $m$ 表示初始未占用的目标节点数量，$n$ 表示瓦片总数。此外，我们证明了智能体可在 $O(n^4)$ 步骤内重构具有孔洞的一大类目标形状。